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C5×D3×D539 #44568391

order32340
automorphism?
homologyAbelianization (H1) = C22×C5 · Schur multiplier (H2) = C2

Aliases

direct productC5×D3×D539
unambiguous semidirect product(D3×C49)⋊(C5×D11)

Splittings

Fitting, HallC3×C5×C11×C49 ⋊ C22
derived, HallC3×C11×C49 ⋊ C22×C5
HallC5×C11×C49 ⋊ C2×D3
HallC3×C5×C49 ⋊ C2×D11
HallC11×C49 ⋊ C2×C5×D3
HallC5×C49 ⋊ D3×D11
HallC3×C5×C11 ⋊ C2×D49
HallC3×C49 ⋊ C2×C5×D11
HallC5×C11 ⋊ D3×D49
SylowC49 ⋊ C5×D3×D11
HallC3×C11 ⋊ C2×C5×D49
HallC3×C5 ⋊ C2×D539
SylowC11 ⋊ C5×D3×D49
SylowC3 ⋊ C2×C5×D539

Subgroups

subgroupG/H
center, hypercenterC5D3×D539
commutator, cosocle, hypocenterC3×C11×C49C22×C5
FrattiniC7C5×D3×D77
FittingC3×C5×C11×C49C22
socleC3×C5×C7×C11C2×D7
LCSG ▷ C3×C11×C49
UCSC1 ◁ C5

120 conjugacy classes of subgroups; only the 44 normal subgroups are shown.

normal subgroups only

This group has 120 conjugacy classes of subgroups — too many to draw. Only its 44 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degreeμ(G) = 68
transitive degreeμt(G) = 8085

A minimal faithful permutation representation — 2 generators on 68 points:

(2,3)(4,5)(6,7)(8,9)(10,11)(13,15)(14,18)(16,21)(17,24)(19,27)(20,26)(22,31)(23,34)(25,36)(28,40)(29,39)(30,38)(32,33)(35,45)(37,47)(41,48)(42,50)(43,49)(44,53)(46,55)(51,56)(52,57)(54,59)(58,60)(61,62,63)(64,65,66,67,68);(1,2,4,6,8,10,11,9,7,5,3)(12,48,23,57,43,45,29,18,37,16,60,31,55,20,27,25,38,54,32,51,15,40,17,50,44,53,42,24,28,13,56,33,59,30,36,19,26,46,22,58,21,47,14,39,35,49,52,34,41)(62,63)

Conjugacy classes & character table

4065 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32340, 93);